Houston Journal of Mathematics. c 2015 University of Houston Volume 41, No. 4, 2015

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1 Houston Journal of Mathematics c 25 University of Houston Volume 4, No. 4, 25 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY TONG TANG, YIFEI PAN, AND MEI WANG Communicated by Min Ru Abstract. We construct a smooth, strictly increasing function with its second derivative changing signs infinitely many times near a point, then we extend the function to a surface increasing in radial direction with curvatures changing signs infinitely often near the origin. The interesting analytical properties of the functions may serve as examples to understand the reversed Hopf Lemma. It is well known that there are many smooth functions vanishing at a point with infinite order. A good example is the function ft = e 2 with f k = lim t f k t = for all k. In this study note, we are interested in finding a strictly increasing function with second derivative changing signs infinitely often near a point. Such function could be useful in understanding the reversed Hopf Lemma, as studied in [] and [2]. To our best knowledge, this example does not seem to be presented in the current circulating publications. Hence in this note, we construct such a function along with its extension to a two dimensional surface strictly increasing in radial direction with curvature changing signs infinitely often near the origin. Example.. Consider the function ut = t. An example t se α/s [ sin /s + C sin 2 /s ] ds, t [, ], α, C >. The function u has the following properties: 2 Mathematics Subject Classification. 26A2, 26A48, 26A5, 26B25. Key words and phrases. Hopf Lemma. 53

2 54 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY Vanishing at t = of infinite order: u k = lim t + u k t = for k =,, 2,. 2 Smoothness of infinite order: u C [, ]. 3 Changing convexity infinitely often at : u changes signs in, ɛ, for any < ɛ <. 4 Strictly increasing: u t > for t, ] when C C α = 4e απ 9e απ /π. Note: C α is an increasing function of α. In particular, C α > 32/π. Remarks: It is more straightforward to construct a function satisfying the first three properties. The main effort is to maintain monotonicity in property 4. Proof of the properties in Example.. By the general Leibnitz rule in calculus, d bt bt ft, sds = b tft, bt a d tft, at + ft, sds, dt at at dt we have u t = t e α/s [ sin /s + C sin 2 /s ] ds, t [, ], u t = e α/t [ sin + C sin 2 ], t, ]. The kth derivative k 2 can be written as u k t = t k e α/t Ht, t, ], where H k t = Ot as t. That is, H k t are uniformly bounded for all t, ], k 2, according to Taylor s Theorem. To show Property, use the fact that for any s R, t s e α/t as t, so lim t + u k t = for all k by the boundedness of sine functions. By the infinite differentiability of e α/t and sin on t >, u C, ]. Property warrants the extension to C [, ]. Thus Property 2 holds. To prove Property 3, we show that for any ɛ,, there exist t, t 2, ɛ with u t >, u t 2 <. Choose n such that 2nπ > /ɛ. Then t = 2nπ + π/2, ɛ, and u t = e α2nπ+π/2 + C >.

3 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY 55 Select δ, ɛ such that C sinδ <. Then t 2 = 2n + π + δ, ɛ, and u t 2 = e α2n+π+δ [ sinπ + δ + C sin 2 π + δ ] Therefore Property 3 holds. = e α2n+π+δ sinδ[ + C sinδ] <. To prove Property 4, for fixed α > and for any t, ], we write u t = t u xdx = u /y dy = sin y + C sin 2 y dy. The above integral converges absolutely for any t >. Express the integral form of u t as u t = I t + In, n=n o with 2no+π I t = C sin 2 y + sin y dy, and n o = n o t = = min{n Z, 2n + π }, 2n+π In = 2n+3π 2n+π sin y + C sin 2 y dy. For t, ], we prove u t > by showing In > for n =,, and I t >. Write 2n+3π 2n+3π In = C sin 2 y dy sin y dy = I n I 2n. 2n+π For the term I n, Shift the intervals to, π and use the monotone property of and y 2, 2n+2π 2n+3π I n = C + sin 2 y dy 2n+π 2n+2π π = C α2n+2π e απ y + 2n + π + sin 2 y dy 2 y + 2n + 2π 2 > C e α2n+2π π > C e α2n+2π π = C e α2n+2π 2n + 3π 2 + e απ π 2. e απ + y + 2n + 2π 2 sin2 y dy e απ e απ + π + 2n + 2π 2 sin2 y dy

4 56 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY Similarly for the second term, 2n+2π I 2 n = + 2n+π 2n+3π 2n+2π sin y dy π = α2n+2π e απ y + 2n + π 2 y + 2n + 2π 2 π < e α2n+2π e απ 2n + π 2 π + 2n + 2π 2 απ = e α2n+2π 2n + 3 2n + 3π 2 2 2n + 2 eaπ sin y dy < e α2n+2π 2n + 3π 2 aπ sin y dy sin y dy 9e απ sin y dy = e α2n+2π 2n + 3π 2 9eαπ 2. Therefore if we choose C C α = 49eαπ πe απ > 49eαπ πe απ +, In = I n I 2 n > e α2n+2π 2n + 3π 2 [ C π + ] e απ 29e απ >. 2 Now consider the boundary interval I t = 2no+π C sin 2 y + sin y dy. If [2n o π, 2n o + π] for some n o, all integrands on the interval of integration so I t >. If [2n o π, 2n o π] for some n o >, notice that C sin 2 y + sin y = only when sin y = /C, so C sin 2 y + sin y < only on two sub-intervals 2no π, 2n o π + sin /C and 2no π sin /C, 2n o π So it is only necessary to consider the worst cases when is the left endpoint of either subintervals. When = 2n o π, the integral I t is over [2n o π, 2n o + π], an interval of length 2π. Then I t = In o > based on the above estimates for In. Now we only need to consider the case when =

5 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY 57 2n o π sin /C, then 2noπ I t = + > C > C > C 2no+π 2n oπ 2no+π 2n oπ 2no+π 2n oπ C sin2 y + sin y dy [ 2noπ sin 2 y dy sin 2 y dy I 2 n o e α2noπ 2n o + π 2 e απ π 2 + 2n o π 2no+π 2n oπ e α2noπ 2n o + π 2 9eαπ 2. sin y dy In the last inequality we use similar derivations used earlier for I 2 n and I n. Therefore I t > when C C α = 4e απ 9e απ /π. Consequently u t = I t + n=n ot In > t, ], ] which proves Property Functions on the plane Next we consider the two dimensional case, where curvature is determined by both the first and second derivatives of the function. Let D R = {x, y R 2, x, y = x2 + < R}. As before, we use D = D for simplicity. Define f : D R by fx, y = ux 2 + for any u C 2, ]. Recall the Gaussian curvature of f κ = κx, y = f xxf yy f 2 xy + f 2 x + f 2 y 2, where f xx = f xx x, y is the second order partial derivative of fx, y with respect to x, and so on. Let s = x 2 +. The following proposition describes the behavior of the curvature near isolated critical points of the radial function f. Proposition 2.. Assume u s =, s,, and u s for s s δ, s + δ \ {s } for some δ, s. Then the Gaussian curvature of fx, y = ux 2 + changes signs in a neighborhood of x, y for any x, y = s cos θ, s sin θ, θ [, 2π.

6 58 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY Proof. The partial derivatives of f can be expressed as f x = 2xu s, f xx = 2u s + 4x 2 u s, f yy = 2u s + 4 u s, f xy = 4xyu s. The denominator of κ is always >. The denominator of κ can be written in the forms 2. Dx, y = f xx f yy fx = 4u s 2 + 8su su s = 4 d su s 2 = Ds. ds For any s s, s + δ, s s Dsds = 4 s s d su s 2 s ds = 4su s 2 = 4s u s 2 >. ds s Therefore there must be t s, s such that Dt >. Consequently for x, y = t cos θ, t sin θ, we have κx, y >. Similarly, for any s 2 s δ, s, s s Dsds = 4su s 2 = 4s 2 u s 2 2 <. s 2 s 2 So there must be t 2 s 2, s such that Dt 2 <. Hence κx 2, < for x 2, = t 2 cos θ, t 2 sin θ. Since δ > can be arbitrarily small, and both x, y, x 2, are in the disk of radius δ centered at x, y, we have proved that κx, y changes signs in a neighborhood of x, y for any x, y = s cos θ, s sin θ, θ [, 2π. The following example is an extension of the function u in Example. to the two dimensional case. Example 2.2. For α >, let v : D R be vx, y = ux 2 +, where u is the function in Example. with C > C α. It is clear that vx, y is strictly increasing in radial directions. Furthermore, the curvature of vx, y changes signs infinite often near, : for any ɛ,, there are x, y, x 2, D ɛ such that κx, y >, κx 2, <. Remarks: Similar to Example., it is not difficult to obtain a surface function with infinitely changing curvature. Thus the main attribute of the constructed function is having infinitely changing curvature while strictly increasing in radial directions. Proof of the curvature property for Example 2.2. ɛ,. Let t = x 2 + and Dt = 4u t 2 + 8tu tu t Fix α, C > C α and

7 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY 59 as defined in 2.. It is sufficient to show that, for ɛ,, there exist t +, t, ɛ with Dt + >, Dt <. To find Dt + > with t +, ɛ is straightforward. Choose n such that 2nπ > /ɛ. Then t + = 2nπ+π/2, ɛ and sin + =, thus u t + = e α2nπ+π/2 + C > Dt + = 4u t t + u t + u t + >. To attain Dt <, we need to choose t, ɛ such that u t < u t 2t then consequently Dt = 8t u t u t + u t such that < C sinδ <. Define ε o = C sinδ[ C sinδ]. 2t Next we examine a u related term Jt to obtain an upper bound. < Jt = e α/t u t t = eα/t t = eα/t t t u xdx = eα/t t <. Select δ, ɛ u /y dy sin y + C sin2 y dy = J t + CJ 2 t Consider the first term J t. Notice that for any t n such that n 2n + π, 2n + 2π, n >, we have n sin y < y dy < sin y y dy = π 2. Therefore for such t n, sin y n y dy = π 2 n We may choose n > N such that sin y y dy as n, t n. J t n = eα/tn sin y t n n y dy = e αy n sin y 2 n t n y d e αy n y sin y n y dy = e αy n sin y dy since 2 n y n y sin y n y dy εo since e αy n C

8 6 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY Now look at the second term CJ 2 t in the u related term Jt. For t n chosen above, < J 2 t n = eα/tn t n n e αy sin 2 y dy = 2 y e αy n y sin n dy = e αz dz = t n n α e αy n n t n sin 2 y dy e αz sin2 z + n dz z + n Choose N 2 such that 2N o+π < ε o α C 2. Then for n N 2, we may choose t n = 2n+π+δ, where δ, ɛ, was selected earlier and was used to define ɛ o. Notice that such t n satisfies the condition n 2n + π, 2n + 2π. For the selected sequence {t n }, we have J 2 t n t n α < α2n + π α2n o + π < ε o C 2. Combining the results we have obtained, for t n = 2n+π+δ and n min{n, N 2 }, we have e α/tn u t n t n = Jt n = J t n + CJ 2 t n < ε o C + C ε o C 2 = 2ε o C. For the u term we can select n max{n, N 2 } such that t n = 2n+π+δ, ɛ, then u t n = e α/tn [ sin n + C sin 2 n ] Then = e α/tn [ sin2n + π + δ + C sin 2 2n + π + δ ] = e α/tn [ sinπ + δ + C sin 2 π + δ ] = e α/tn sinδ[ + C sinδ] = e α/tn C sinδ[ C sinδ] C = e α/tn ε o C. e α/tn u t n = ε o C < 2 eα/tn u t n t n = u t n < u t n 2t n Let t = t n for any of the n max{n, N 2 }. Then Dt < with t, ɛ. This proves the claimed curvature property in Example 2.2.

9 AN INCREASING FUNCTION WITH INFINITELY CHANGING CONVEXITY 6 References [] Li, Y. and Nirenberg, L. 25 A geometric problem and the Hopf Lemma. I. J. Eur. Math. Soc. 8 26, [2] Li, Y. and Nirenberg, L. 26A geometric problem and the Hopf Lemma. II. Chin. Ann. Math. 27B2, 26, Received April 8, 23 Tong Tang, Department of Mathematics, Nanjing Normal University, Nanjing 297, China address: tt5756@26.com Yifei Pan, Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, Fort Wayne, IN School of Mathematics and Informatics, Jiangxi Normal University, Nanchang, China address: pan@ipfw.edu Mei Wang, Department of Statistics, University of Chicago, Chicago, IL address: meiwang@galton.uchicago.edu